Displays of older iPods (Video) tend to break after some years, at least the 2 present in our home did. The back-light becomes disfunct in an horizontal area in the middle of the screen. The broken area is irregular, It's not just a horizontal bar, but an band of varying width.
Smooth following the description from iFixit. After re-assembly there was some flickering of the screen, caused a bad contact of the display cable. I disconnected the display cable - it looked like it was clamped partially underneath the display - and was then able to slide it into the connector completely. After second re-assembly the display was functioning properly again.
I will now go and give the second iPod a go.